Season chicken with salt and pepper. Dredge in flour, shaking off the excess. Heat oil in a large deep skillet or Dutch oven. Add chicken; cook until browned, about 4 minutes per side. Transfer to a plate.
Add shallots to the pan; cook, stirring, until fragrant, about 30 seconds. Add wine and scrape up any browned bits. Simmer until reduced slightly, about 3 minutes. Add broth; bring to a simmer. Return the chicken to the pan; add carrot, mushrooms and tarragon sprigs. Reduce heat to low, cover and simmer gently until the chicken is tender and no longer pink in the center, about 20 minutes. Transfer the chicken to a plate; cover with foil to keep warm. Discard tarragon sprigs. Increase heat to medium-high. Simmer the cooking liquid for 2 to 3 minutes to intensify flavor. Add cornstarch mixture and cook, stirring, until slightly thickened, about 2 minutes. Whisk in sour cream, mustard and chopped tarragon.